On the maximum curvature of closed curves in negatively curved manifolds

Motivated by Almgren's work on the isoperimetric inequality, we prove a sharp inequality relating the length andmaximumcurvature of a closed curve in a complete, simply connectedmanifold of sectional curvature at most -1. Moreover, if equality holds, then the norm of the geodesic curvature is constant and the torsion vanishes. The proof involves an application of the maximum principle to a function deûned on pairs of points.